Magnetic-free non-reciprocal devices exhibiting non-reciprocity through angular momentum biasing

ABSTRACT

A non-reciprocal device incorporating metamaterials which exhibit non-reciprocity through angular momentum biasing. The metamaterial, such as a ring resonator, is angular-momentum biased. This is achieved by applying a suitable mechanical or spatio-temporal modulation to resonant inclusions of the metamaterial, thereby producing strong non-reciprocity. In this manner, non-reciprocity can be produced without requiring the use of large and bulky magnets to produce a static magnetic field. The metamaterials of the present invention can be realized by semiconducting and/or metallic materials which are widely used in integrated circuit technology, and therefore, contrary to magneto-optical materials, can be easily integrated into the non-reciprocal devices and large microwave or optical systems. The metamaterials of the present invention can be compact at various frequencies due to the enhanced wave-matter interaction in the constituent resonant inclusions. Additionally, by using the metamaterials of the present invention, the power consumed in the biasing network is drastically reduced.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to the following commonly owned U.S. patentapplication:

Provisional Application Ser. No. 61/857,580, “Magnetic-FreeNon-Reciprocal Devices Based on Metamaterials Exhibiting Non-ReciprocityThrough Angular Momentum Biasing,” filed Jul. 23, 2013, and claims thebenefit of its earlier filing date under 35 U.S.C. §119(e).

GOVERNMENT INTERESTS

This invention was made with government support under Grant No.HDTRA1-12-1-0022 awarded by the Defense Threat Reduction Agency; andFA9550-11-1-0009 awarded by the Air Force Office of Scientific Research.The government has certain rights in the invention.

TECHNICAL FIELD

The present invention relates generally to non-reciprocal devices, andmore particularly to magnetic-free non-reciprocal devices exhibitingnon-reciprocity through angular momentum biasing.

BACKGROUND

Non-reciprocal devices, such as isolators, circulators and phaseshifters, are important in electrical systems, such as communicationnetworks, to prevent adverse backward reflection, interference andfeedback. Traditionally, non-reciprocity in microwaves and optics isachieved by using magneto-optical materials, where non-reciprocity isthe result of biasing with a static magnetic field. By requiring the useof a static magnetic field, heavy and bulky magnets are required whichare difficult to integrate in the non-reciprocal devices. As a result,different biasing schemes have been devised in an attempt to eliminatethe use of heavy and bulky magnets.

One such scheme involves the use of metamaterials based ontransistor-loaded ring resonators that are biased via direct current.Unfortunately, such a scheme involves significant power consumption(power consumption in the biasing network and the transistor itself) andits operation is limited to microwave frequencies.

Another scheme involves the use of optical isolators consisting ofspatially-temporarily modulated waveguides that are biased via linearmomentum. Unfortunately, such a scheme is limited to specificapplications where the spatially-temporarily modulated waveguides needto be many wavelengths long. Furthermore, such a scheme also involvessignificant power consumption in the biasing network.

BRIEF SUMMARY

In one embodiment of the present invention, a non-reciprocal devicecomprises a substrate and a metamaterial on the substrate. An angularmomentum is applied to the metamaterial thereby producingnon-reciprocity.

In another embodiment of the present invention, a non-reciprocal devicecomprises a substrate and a periodically arranged pair of metallic ringspatterned on both sides of a dielectric layer, where the pair ofmetallic rings is formed on the substrate and where the pair of metallicrings is angular-momentum biased. Furthermore, the pair of metallicrings is permittivity modulated by loading the rings with time-variablecapacitors at equidistant azimuthal positions thereby producingnon-reciprocity.

In another embodiment of the present invention, a non-reciprocal devicecomprises a substrate and an angular-momentum biased optical ringresonator on the substrate, where the optical ring resonator ispermittivity modulated.

In a further embodiment of the present invention, a non-reciprocaldevice comprises a ring of three resonators symmetrically coupledtogether, where each of the resonators is phase shifted 120° withrespect to each other and operable at either microwave, light or soundwaves.

In another embodiment of the present invention, a non-reciprocal devicecomprises a plurality of resonators, where each of the plurality ofresonators comprises a pair of capacitors where a capacitance of thepair of capacitors is equally distributed at both sides of one or moreinductors connected in series to the pair of capacitors. Furthermore,each of the plurality of resonators is modulated via capacitancemodulation. The non-reciprocal device further comprises a plurality oftransmission lines coupled to the plurality of resonators.

In a further embodiment of the present invention, a non-reciprocaldevice comprises a photonic crystal comprising an array of dielectricrods. The non-reciprocal device further comprises three gaps in thearray of dielectric rods, where light is localized in each of the threegaps in a frequency range for the gap thereby causing each of the threegaps to function as a resonator.

The foregoing has outlined rather generally the features and technicaladvantages of one or more embodiments of the present invention in orderthat the detailed description of the present invention that follows maybe better understood. Additional features and advantages of the presentinvention will be described hereinafter which may form the subject ofthe claims of the present invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description is considered in conjunction with thefollowing drawings, in which:

FIG. 1A illustrates the geometry of an azimuthally symmetric ringresonator with spatio-temporal (ST) modulated permittivity in accordancewith an embodiment of the present invention;

FIGS. 1B and 1C are graphs illustrating the transformation of the stateswith angular momentum ±1 in the frequency/angular momentum plane for themodulation orbital angular momentum being 1 and 2, respectively, inaccordance with an embodiment of the present invention;

FIG. 1D is a frequency diagram of the eigen-states of the ring resonatorof FIG. 1A without and with spatio-temporal modulation for themodulation orbital angular momentum being 2 in accordance with anembodiment of the present invention;

FIG. 1E is a graph illustrating the substate eigen-frequencies versusthe modulation frequency for the modulation orbital angular momentum(L_(m)) being 2 in accordance with an embodiment of the presentinvention;

FIG. 1F is a graph illustrating the substate energies versus themodulation frequency for L_(m)=2 in accordance with an embodiment of thepresent invention;

FIG. 2A illustrates a spatio-temporal (ST) modulated metasurfaceconsisting of periodically arranged pairs of broadside-parallel metallicrings patterned on both sides of a thin dielectric layer in accordancewith an embodiment of the present invention;

FIG. 2B illustrates an implementation of capacitance modulation inaccordance with an embodiment of the present invention;

FIG. 2C is a graph illustrating the transmission through the unmodulatedmetasurface in accordance with an embodiment of the present invention;

FIG. 2D is a graph illustrating transmission in the +z direction with acapacitance modulation of 0.02 pF and a modulation frequency of 0.1 GHzin accordance with an embodiment of the present invention;

FIG. 2E is a graph illustrating transmission in the +z direction with acapacitance modulation of 0.02 pF and a modulation frequency of 0.5 GHzin accordance with an embodiment of the present invention;

FIG. 3A is a graph illustrating the polarization rotation angle fordifferent modulation frequencies and a capacitance modulation of 0.02 pFin accordance with an embodiment of the present invention;

FIG. 3B is a graph illustrating the corresponding ellipticity angle fordifferent modulation frequencies and a capacitance modulation of 0.02 pFin accordance with an embodiment of the present invention;

FIG. 4A is a graph illustrating transmission versus frequency for anoptical channel-drop filter in accordance with an embodiment of thepresent invention;

FIG. 4B is a graph illustrating non-reciprocal transmission versusfrequency for the optical channel-drop filter under spatiotemporalmodulation in accordance with an embodiment of the present invention;

FIG. 5A illustrates a magnetically biased three-port junction, resultingin a non-reciprocal scattering response in accordance with an embodimentof the present invention;

FIGS. 5B(1)-5B(2) illustrate an angular-momentum-biased ring circulatorand spatiotemporally-modulated lumped circuit circulator in accordancewith an embodiment of the present invention;

FIG. 6 is a graph of the simulated results for the magnitude of thetransmission coefficients in a microstrip resonator designed followingthe same principle as in FIG. 9 in accordance with an embodiment of thepresent invention;

FIG. 7A illustrates a ring structure consisting of three stronglycoupled identical and symmetrically-coupled resonant tanks withresonance frequency ω₀ and coupling factor κ in accordance with anembodiment of the present invention;

FIG. 7B is a frequency diagram of the hybrid states |R

and |L

of the modulated ring versus ω_(m) in accordance with an embodiment ofthe present invention;

FIG. 8A illustrates the constituent resonator of the ring: an L-C tankwith modulated capacitance in accordance with an embodiment of thepresent invention;

FIG. 8B illustrates a ring formed by three identical resonators coupledthrough three identical capacitances C_(c) in accordance with anembodiment of the present invention;

FIG. 8C is a photograph of the fabricated prototype of the ring of FIG.8B in accordance with an embodiment of the present invention;

FIG. 9 illustrates the physical layout of the RF non-reciprocalcoupled-resonator ring in accordance with an embodiment of the presentinvention;

FIG. 10 is a table of the values of the lumped elements used in thefabricated layout in accordance with an embodiment of the presentinvention;

FIG. 11 illustrates the experimental setup in accordance with anembodiment of the present invention;

FIG. 12 is a table of the equipment used during the measurement of thering in accordance with an embodiment of the present invention;

FIG. 13A is a graph illustrating the measured transmission from port 1to ports 2 and 3 for no modulation (V_(m)=0 V), where the power isequally split to the output ports (ports 2 and 3) in accordance with anembodiment of the present invention;

FIG. 13B is a graph illustrating the measured scattering parameters whenV_(m)=0.6 V in accordance with an embodiment of the present invention;

FIG. 13C presents the S-parameters obtained through combined full-waveand circuit simulations in accordance with an embodiment of the presentinvention;

FIG. 14A is a graph of the measured and simulated transmission betweenports 1 and 2 in accordance with an embodiment of the present invention;

FIG. 14B is a graph of the isolation (S₁₂/S₂₁) in a logarithmic scale inaccordance with an embodiment of the present invention;

FIG. 15 shows the measured isolation versus frequency for V_(dc) variedbetween 1.73 V and 4.5 V in a logarithmic scale in accordance with anembodiment of the present invention; and

FIG. 16 illustrates a photonic crystal used for building resonators forlight in accordance with an embodiment of the present invention.

DETAILED DESCRIPTION

As stated in the Background section, non-reciprocal devices, such asisolators, circulators and phase shifters, are important in electricalsystems, such as communication networks, to prevent adverse backwardreflection, interference and feedback. Traditionally, non-reciprocity inmicrowaves and optics is achieved via magneto-optical materials, wherenon-reciprocity is the result of biasing with a static magnetic field.By requiring the use of a static magnetic field, heavy and bulky magnetsare required which are difficult to integrate in the non-reciprocaldevices. As a result, different biasing schemes have been devised in anattempt to eliminate the use of heavy and bulky magnets. One such schemeinvolves the use of metamaterials based on transistor-loaded ringresonators that are biased via direct current. Unfortunately, such ascheme involves significant power consumption (power consumption in thebiasing network) whose operation is limited to microwave frequencies.Another scheme involves the use of optical isolators consisting ofspatially-temporarily modulated waveguides that are biased via linearmomentum. Unfortunately, such a scheme is limited to specificapplications where the spatially-temporarily modulated waveguides needto be many wavelengths long. Furthermore, such a scheme also involvessignificant power consumption in the biasing network.

The principles of the present invention provide a means for overcomingthese deficiencies by developing a novel class of metamaterials thatexhibit non-reciprocity through angular momentum biasing. Theconstituent element of these metamaterials is an azimuthally spatiallyand temporally modulated ring resonator. As a result, non-reciprocitycan be produced with a low frequency electrical signal in contrast tomagneto-optical materials requiring the use of large and bulky magnetsto produce a static magnetic field. Furthermore, the metamaterials ofthe present invention can be realized by semiconducting and/or metallicmaterials, such as silicon and gold, which are widely used in integratedcircuit technology, and therefore, contrary to magneto-opticalmaterials, can be easily integrated into the non-reciprocal devices.Additionally, the metamaterials of the present invention can be compactat various frequencies (e.g., microwave, optical) due to the enhancedlight-matter interaction in the constituent resonant inclusions. Incontrast, magneto-optical devices can be bulky at optical frequenciesdue to the weak magneto-optical effect at these frequencies. Inaddition, the metamaterials of the present invention can be implementedat a wide range of frequencies, such as from microwave to light, incontrast to transistor-based metamaterials whose operation is limited tomicrowave frequencies. Furthermore, the metamaterials of the presentinvention can be more compact due to the enhanced light-matterinteraction of their resonant inclusions (elements) as opposed tospatially-temporarily modulated waveguides which need to be manywavelengths long.

Additionally, by using the metamaterials of the present invention, thepower consumed in the biasing network is drastically reduced. Adiscussion of non-reciprocal devices implementing metamaterialsexhibiting non-reciprocity through angular momentum biasing is providedbelow in connection with FIGS. 1A-1F, 2A-2E, 3A-3B and 4A-4B. FIG. 1Aillustrates the geometry of an azimuthally symmetric ring resonator withspatio-temporal (ST) modulated permittivity. FIGS. 1B and 1C are graphsillustrating the transformation of the states with angular momentum ±1in the frequency/angular momentum plane for the modulation orbitalangular momentum being 1 and 2, respectively. FIG. 1D is a frequencydiagram of the eigen-states of the ring resonator of FIG. 1A without andwith spatio-temporal modulation for the modulation orbital angularmomentum being 2. FIG. 1E is a graph illustrating the substateeigen-frequencies versus the modulation frequency for the modulationorbital angular momentum (L_(m)) being 2. FIG. 1F is a graphillustrating the substate energies versus the modulation frequency forL_(m)=2. FIG. 2A illustrates a spatio-temporal (ST) modulatedmetasurface consisting of periodically arranged pairs ofbroadside-parallel metallic rings patterned on both sides of a thindielectric layer. FIG. 2B illustrates an implementation of capacitancemodulation. FIG. 2C is a graph illustrating the transmission through theunmodulated metasurface. FIG. 2D is a graph illustrating transmission inthe +z direction with a capacitance modulation of 0.02 pF and amodulation frequency of 0.1 GHz. FIG. 2E is a graph illustratingtransmission in the +z direction with a capacitance modulation of 0.02pF and a modulation frequency of 0.5 GHz. FIG. 3A is a graphillustrating the polarization rotation angle for different modulationfrequencies and a capacitance modulation of 0.02 pF. FIG. 3B is a graphillustrating the corresponding ellipticity angle for differentmodulation frequencies and a capacitance modulation of 0.02 pF. FIG. 4Ais a graph illustrating transmission versus frequency for an opticalchannel-drop filter. FIG. 4B is a graph illustrating non-reciprocaltransmission versus frequency for the optical channel-drop filter underspatiotemporal modulation.

Referring now to the Figures in detail, a possible constituent inclusion(element) of the metamaterial of the present invention is a simple ringresonator 101 (may also be a collection of ring resonators) on asubstate 102 loaded with a spatio-temporal (ST) azimuthally modulatedpermittivity as schematically shown in FIG. 1A in accordance with anembodiment of the present invention, where φ is the azimuthal coordinatein a cylindrical reference system co-centered with the inclusion andL_(m) is the modulation orbital angular momentum. In absence ofmodulation, Δ∈_(m)=0, ring 101 supports degenerate counter-propagatingstates |±l

e^(−iω) ^(l) ^(t) with azimuthal dependence e^(±ilφ), resonating whenthe ring circumference is l times the guided wavelength, which impliesthat, for the fundamental |±1

states used herein, the ring dimensions are smaller than the wavelength.The resonant size can be further reduced by adding capacitances alongthe loop, as in split-ring resonator design. As will be shown shortly,introducing suitable ST azimuthal modulation can lift the degeneracybetween the |±1

states and produce non-reciprocity, an effect that can be interpreted asthe metamaterial analog of a static magnetic bias removing thedegeneracy between atomic states of opposite orbital angular momenta inmagnetic materials.

The proposed permittivity modulation is a type of amplitude modulation,and, as such, it results in the generation of two intermodulationproducts |k+L_(m)

e^(−i(ω) ^(k) ^(+ω) ^(m) ^()t) and |k−L_(m)

e^(−i(ω) ^(k) ^(−ω) ^(m) ^()t) for each state |±k

e^(−iω) ^(k) ^(t). If any of these products overlap in frequency withanother state |±l

e^(−iω) ^(l) ^(t), resonant coupling between the |k

and |t

states occurs, significantly affecting both resonances. Since the goalof the present invention is to lift the degeneracy between |±l

states, L_(m)=1 and ωm=ω₂−ω₁ might appear the most reasonable choice, sothat the |+1

state gets resonantly coupled to the |+2

state, while no coupling occurs for the |−1

state, as illustrated in FIG. 1B. FIGS. 1B and 1C are graphsillustrating the transformation of the states with angular momentum ±1in the frequency/angular momentum plane for the modulation orbitalangular momentum being 1 and 2, respectively, in accordance with anembodiment of the present invention. Specifically. FIGS. 1B and 1Cillustrate the transformation of the |±1

states in the frequency and angular momentum plane for L_(m)=1 andL_(m)=2, respectively. Referring to FIG. 1B, ω₂ is usually close to 2ω₁,and, as a result, ω_(m) should be close to ω₁, which may be challengingto achieve especially at terahertz and optical frequencies. For L_(m)=2,in contrast, the states |±1

resonantly couple to each other for ω_(m)=0, as illustrated in FIG. 1C.For ω_(m) equaling zero, the structure is obviously reciprocal, but anysmall departure from zero can break reciprocity, and, no matter howlarge ω₁ is, strong nonreciprocal response may be obtained by properlychoosing Δ∈_(m) and the resonator Q-factor, as will be discussed in moredetail in the following.

This concept may be analyzed using coupled-mode theory: the amplitudesa_(±1) of the |±1

states satisfy the equations:

$\begin{matrix}{{{\overset{.}{a}}_{+ 1} = {{{- {\mathbb{i}\omega}_{1}}a_{+ 1}} + {{\mathbb{i}}\frac{1}{2}\omega_{1}\kappa_{m}{\mathbb{e}}^{{- {\mathbb{i}\omega}_{m}}t}a_{- 1}}}},{{\overset{.}{a}}_{- 1} = {{{- {\mathbb{i}\omega}_{1}}a_{- 1}} + {{\mathbb{i}}\frac{1}{2}\omega_{1}\kappa_{m}{\mathbb{e}}^{{\mathbb{i}\omega}_{m}t}a_{+ 1}}}},} & (1)\end{matrix}$whereκ_(m)=2π∫_(S) _(t) Δ∈_(m) |E _(t1)|² ρdρdz  (2)is the coupling coefficient between the |±1

states, with S_(t) being the modal cross-section of the resonator andE_(t1) the corresponding normalized electric field distribution. Forhomogeneous resonators (Δ∈_(m) and ∈ uniform, where ∈ is the backgroundpermittivity), κ_(m)=Δ∈_(m)/2∈. The solution of Equation (1) providesthe eigen-states of the modulated ring:

$\begin{matrix}{{\left. {{\left. {\left. {{\left. {{\left. {\left. \left| \alpha \right. \right\rangle = \left| {+ 1} \right.} \right\rangle{\mathbb{e}}^{{\mathbb{i}\omega}_{\alpha}t}} + \frac{\Delta\omega}{\omega_{1}\kappa_{m}}} \middle| {- 1} \right\rangle{\mathbb{e}}^{{- {{\mathbb{i}}{({\omega_{\alpha} - \omega_{m}})}}}t}},\left| \beta \right.} \right\rangle = \left| {- 1} \right.} \right\rangle{\mathbb{e}}^{{\mathbb{i}\omega}_{\beta}t}} - \frac{\Delta\omega}{\omega_{1}\kappa_{m}}} \middle| {+ 1} \right\rangle{\mathbb{e}}^{{- {{\mathbb{i}}{({\omega_{\beta} + \omega_{m}})}}}t}},} & (3)\end{matrix}$where ω_(a)=ω₁−Δω/2, ω_(β)=ω₁+Δω/2 and Δω=√{square root over (ω_(m) ²+ω₁²κ_(m) ²)}−ω_(m). This solution may be extended to take into account thepresence of loss and coupling with the excitation signals.

The states |α

and |β

are hybridizations of the non-modulated ring states |±1

and |−1

, which are generally characterized by different frequencies andamplitudes, as illustrated in FIG. 1D. FIG. 1D is a frequency diagram ofthe eigen-states of the ring resonator of FIG. 1A without and withspatio-temporal modulation for L_(m)=2. In the absence of modulation(ω_(m)=0), the sub-states |±1

of each hybrid state share the same frequency and energy, as shown inFIGS. 1E and 1F, respectively, and the system is reciprocal as expected.FIG. 1E is a graph illustrating the substate eigen-frequencies versusthe modulation frequency for L_(m)=2 in accordance with an embodiment ofthe present invention. FIG. 1F is a graph illustrating the substateenergies versus the modulation frequency for L=2 in accordance with anembodiment of the present invention. However, when modulation isintroduced (ωm≠0), the sub-states split (see FIG. 1E) andnon-reciprocity arises. This splitting follows from the simultaneousspatial and temporal nature of the proposed modulation, generating thestates |k+2

e^(−i(ω+ω) ^(m) ^()t) and |k−2

e^(−i(ω+ω) ^(m) ^()t) from |k

e−^(iωt) (see FIG. 1C). Therefore, if the sub-state |+1

exists at frequency ω, the sub-state |−1

can only exist at frequency ω−ω_(m). For ω_(m)≠0 the energy is unevenlydistributed between sub-states (see FIG. 1F), with the unbalanceincreasing with ω_(m). The dominant sub-states for |α

and |β

are |+1

and |−1

, respectively, as indicated in FIGS. 1D-1F (see lines 103-107). Thesecondary sub-states are indicated by lines 108-112 in FIGS. 1D-1F.

The amount of non-reciprocity is determined by the minimum distancebetween sub-states of opposite handedness Δω_(min)=min{ω_(m),ω_(β)−ω_(α)=Δω} and by the resonance width ω₁/Q, where Q is theresonance quality factor, corresponding to the inverse of the fractionalbandwidth. In practical devices, such as polarization rotators andcirculators, which are based on interference between states, Δω_(min)and ω₁/Q is of the same order. It can be proven from Equation (3) thatΔω_(min)≦ω₁κ_(m)/√{square root over (3)}, with the maximum value holdingfor ω_(m)=Δω=ω₁κ_(m)/√{square root over (3)}. Therefore Qκ_(m)=√{squareroot over (3)}, consistently with the expectation that a lower Qresonator requires a higher κ_(m) and subsequently a higher Δ∈_(m).

Consider now an ST-modulated metasurface 201 consisting of periodicallyarranged pairs of broadside-parallel metallic rings 202 patterned onboth sides of a thin dielectric layer 203, as shown in FIG. 2A inaccordance with an embodiment of the present invention. In oneembodiment, metal rings 202 are formed on a substrate 204. Permittivitymodulation is effectively obtained by loading the rings withtime-variable capacitors ΔC_(n)=ΔC_(m) cos(ω_(m)t−2φ_(n)) at equidistantazimuthal positions φ_(n)=nπ/4, where n=0 . . . 7, which is equivalentto applying a continuous capacitance modulationΔC=(4ΔC_(m)/π)cos(ω_(m)t−2φ). Notice that the modulation amplitude4ΔC_(m)/π is the average of the localized capacitance ΔC_(m) over thediscretization period π/4, as may be intuitively expected. A possiblepractical implementation of this capacitance modulation is illustratedin FIG. 2B in accordance with an embodiment of the present invention: itconsists of a varactor 205, as its core element, a DC biasing source206, an AC modulation source 207 with frequency ω_(m) and appropriatefilters (band-stop filter (BSF) 208 and band-pass filter (BPF) 209) thatminimize the interference between ring 202 and the biasing network. Sucha circuit may be easily integrated into the ring substrate 204 (FIG. 2A)within conventional printed circuit technology. Furthermore, sincevaractors and filters are low-loss components, the overall powerconsumption is expected to be very low.

In the absence of modulation, metasurface 201 exhibits two resonances,shown in the inset of FIG. 2C in accordance with an embodiment of thepresent invention, a low-Q ‘bright’ mode at 23 GHz (see line 210) withparallel currents induced in the two rings 202, and a coupled high-Q‘dark’ mode at 9 GHz (see line 211) with antiparallel currents. Suitablecoupling between these two modes results in a peculiar Fano resonantresponse at 8.9 GHz, with a sharp transition from full to notransmission. This response is ideal for the purpose of the presentinvention, since its sharp frequency response relaxes the requirementson the modulation capacitance, and, at the same time, leads to strongnon-reciprocal effects because of the associated antiparallel currentsin the rings, maximizing the excitation of the modulation capacitors.

FIG. 2D is a graph illustrating the transmission of circularly polarized(CP) waves through metasurface 201 (along +z) for ΔC_(m)=0.02 pF andf_(m)=100 MHz in accordance with an embodiment of the present invention.The chosen value of ΔC_(m) corresponds to an effective capacitancemodulation of 0.026 pF/rad, which, considering the ring-pair capacitance0.48 pF/rad, leads to κ_(m)=0.027. This yields ω₁k_(m)≈0.24 GHz, whichis enough for a clear separation between |α

and |β

states. Small variations of ΔC_(m) that may occur in practice can beeasily compensated by adjusting ω_(m), as long as the |α

and |β

states are distinguishable.

As expected, the response is different for right-handed CP (RHCP) (seeline 212) and left-handed CP (LHCP) excitations (see line 213), and ineach case two resonant dips are observed, with the stronger oneresulting from coupling with the state whose dominant sub-state is ofthe same handedness as the incident wave. RHCP incident waves 212strongly couple to |α

at frequency f_(α), while LHCP waves 213 strongly couple to |β

at f_(β). The weaker resonant dips correspond to the secondarysub-states at f_(β)+f_(m) (RHCP excitation, substate |+1

of |β

) and f_(α)-f_(m) (LHCP waves, |−1

of |α

). If the structure is excited from −z, the transmission curves of FIG.2C switch handedness, as the incident wave feels opposite modulationspin, a clear demonstration of non-reciprocity. This polarizationtransmission asymmetry can be exploited to realize, for instance, a CPisolator by placing the transmission null of one polarization at thesame frequency as the transmission peak of the other polarization. Thiscondition is fulfilled for f_(m)=0.5 GHz, as shown in FIG. 2E inaccordance with an embodiment of the present invention. FIG. 2E is agraph illustrating the transmission of circularly polarized (CP) wavesthrough metasurface 201 (along +z) for ΔC_(m)=0.02 pF and f_(m)=0.5 GHzin accordance with an embodiment of the present invention. Referring toFIG. 2E, at 8.91 GHz, RHCP and LHCP waves (see lines 214, 215,respectively) can penetrate metasurface 201 only from −z and +z,respectively. This operation may be the basis of different types ofpolarization-dependent microwave isolators.

Another important non-reciprocal effect, common in ferromagneticmaterials, is Faraday rotation, i.e., the non-reciprocal rotation of thepolarization plane of a wave, as it propagates through the material. Therotation is opposite for opposite propagation directions, as it isdetermined by the (fixed) bias direction. One can achieve the sameeffect in the proposed ST modulated metasurface 201 (see FIG. 2A), asshown in FIG. 3A, which plots the polarization rotation angle θ fordifferent f_(m) and ΔC_(m)=0.02 pF in accordance with an embodiment ofthe present invention. As the modulation frequency increases, θincreases and the bandwidth decreases. The bandwidth reduction isclearly due to the decrease of Δω as f_(m) increases, but the monotonicincrease of θ may seem contradictory with the fact that the separationbetween states, which determines the amount of non-reciprocity, isactually reduced as f_(m) increases. This peculiar monotonic increase ofθ results from the fact that at resonance the transmission coefficientfor x-polarized waves T_(xx) decreases faster than the transmissioncoefficient of x- to y-polarized waves T_(yx), so that T_(yx)/T_(xx),which is proportional to θ, actually increases. For f_(m)=0.5 GHz,T_(yx) is maximum and θ=60°, corresponding to a giant rotation of 6000°per free-space wavelength, without any magnetic bias. FIG. 3B is a graphillustrating the ellipticity angle χ at the output for linearlypolarized inputs in accordance with an embodiment of the presentinvention. As shown in FIG. 3B, ellipticity angle χ is zero at thefrequency of maximum θ, implying that the transmitted field is linearlypolarized, making the proposed metasurface particularly exciting forapplications requiring non-reciprocal linear polarization rotation.

As previously discussed, the proposed scheme of ST modulation withL_(m)=2 poses no restriction on the modulation frequency, opening thepossibility to apply the proposed concept to optical frequencies. As aproof of concept, an optical isolator based on an ST modulatedchannel-drop filter was designed with the following geometricalparameters of the optical isolator 401 (shown in FIGS. 4A and 4B): R(radius of optical ring resonator)=0.88 a, w (width of channel/dropwaveguide)=0.2 a and g (gap between ring and waveguide)=0.3 a, where ais an arbitrary reference length. For operation with a wavelength of1.55 μm, the corresponding absolute values are a=1.04 μm, R=0.92 μm,w=0.21 μm and f_(m)=60 GHz. The modulation frequency for the case ofFIG. 4B (discussed further below) is f_(m)=2×10⁴ (c/a), where c is thespeed of light. The optical isolator 401 of FIGS. 4A and 4B includes anoptical ring resonator 402, which is between a set of waveguides(channel waveguide 403 with ports 1 and 2 and drop waveguide 404 withports 3 and 4). Optical isolator 401 of FIGS. 4A and 4B is formed on asubstrate 405. When light of a resonant wavelength is passed through theloop from the input waveguide (e.g., waveguide 403), it builds up inintensity over multiple round-trips in ring resonator 402 due toconstructive interference and is output to the output waveguide (e.g.,waveguide 404) which serves as a detector waveguide. The discussion ofthe operation of optical ring resonator 402 will be discussed below inconnection with FIGS. 4A-4B. FIG. 4A is a graph illustratingtransmission versus frequency without modulation for optical ringresonator 402 discussed above in accordance with an embodiment of thepresent invention. FIG. 4B is a graph illustrating non-reciprocaltransmission versus frequency with spatio-temporal (ST) modulationturned on for the optical isolator discussed above in accordance with anembodiment of the present invention.

As shown in FIG. 4A, without modulation, the power entering structurethrough the channel waveguide 403 from either port 1 or 2 couples to theright- or left-handed ring resonance, respectively, creating atransmission dip at resonance. Splitting the ring resonances with properazimuthal ST modulation moves the transmission dips to differentfrequencies for opposite propagation directions, thus creatingnon-reciprocity. The isolator 401 can be realized on silicon (Si), whichexhibits the strongest electro-optic effect observed to date, withtypical values around Δ∈_(m)=5×10⁻⁴, where ∈_(s) is its permittivity,leading to κ_(m)≈2.5×10⁴. Such permittivity modulation can be obtainedusing PIN diodes. According to the bandwidth criterion QC_(m)=√{squareroot over (3)}, a Q-factor ˜7,000 would be sufficient for adequateseparation of the |α

and |β

states. Such level of Q-factor is common in Si-photonics integratedsystems, and in the design of FIG. 4A, it is achieved using the 11-thazimuthal resonance of ring 402. Although the theory above was derivedfor the |±1

states it can be easily extended to any pair of states |±l

by substituting L_(m)=2 with L_(m)=2l. Therefore, L_(m)=22 in the designof optical isolator 401, which may be achieved by uniformly integrating88 PIN diodes along the ring perimeter of ring 402, leading to aseparation of 65 nm between consecutive diodes. It should be noted thatthis design has not been optimized and that a significantly lower numberof PIN diodes may be still sufficient to achieve a similar effect inoptimized geometries.

The simulated scattering parameters of the structure without and withmodulation are presented in FIGS. 4A and 4B, respectively. In theabsence of modulation, S₂₁=S₁₂ (S_(ij) being the transmissioncoefficient from port j to port i), meaning that the system isreciprocal. When the modulation is applied, the right- and left-handedresonances of the ring split and non-reciprocity occurs, as shown inFIG. 4B. For instance, at the right-handed resonance, indicated in FIG.4B with the dashed line, transmission from port 1 to 2 is significantlylower than transmission from port 2 to 1, an effect that can also beseen in the corresponding field plots in the inset. This operation isobtained within a ring structure that is comparable in size to theoperation wavelength λ=1.55 μm, and without the need of magnetic bias.

The principles of the present invention provide a new paradigm toachieve magnetic-free non-reciprocity via angular momentum biasing basedon resonant rings with specifically tailored ST azimuthal modulation.The proposed form of modulation removes the degeneracy between oppositeresonant states, which, combined with suitably induced high-Q response,realizes giant non-reciprocity in subwavelength components with moderatemodulation frequencies and amplitudes. A few applications based on thisapproach have been presented, including an ultrathin radio-frequencyisolator, giant Faraday rotation and an optical isolator, all realizedwithout requiring bulky magnetic biasing elements. The proposed approachopens pathways towards non-reciprocal integrated microwave andnanophotonic components for a variety of applications.

In another embodiment of the present invention, spatiotemporal biasingis used to generate the electrical rotation necessary forangular-momentum biasing as discussed further below. Such an approachadds several benefits, including, but not limited to, being compact,scalable in frequency, fully compatible with complex integratedcircuits, inherently linear and consisting of completely passivecomponents.

When a three port resonator is biased with a magnetic field, as seen inFIG. 5A, the degenerate counter rotating modes are split, leading to thenon-reciprocal response required in a circulator. FIG. 5A illustratesmagnetically biased three-port junction 500, resulting in anon-reciprocal scattering response in accordance with an embodiment ofthe present invention. This approach is generally used to realizeferromagnetic-based circulators. The angular-momentum biasing techniquerelies on electrical rotation, which may be realized in several designs.The spatiotemporal biasing technique may be applied to both microstripand lumped element resonators, as shown in FIGS. 5B(1)-5B(2). FIGS.5B(1)-5B(2) illustrate an angular-momentum-biased ring circulator 501and spatiotemporally-modulated lumped circuit circulator 502 inaccordance with an embodiment of the present invention. Modulation ofthe capacitance of the ring electrically emulates the angular-momentum“spin.” At microwave frequencies, such a modulation is implemented byloading the ring with varactors. Other modulation techniques, such aswith micromechanical capacitors or FET-based variable capacitors, areavailable, but ease of integration and cost considerations led to thechoice of varactors for the microwave frequency designs.

The conditions for maximum isolation in a circulator may be derived fromtemporal coupled mode theory as,f ₁ =f ₂κ₁/√{square root over (3)},Qκ ₁=1,  (4)where f₁ is the modulation frequency, f₂ the carrier frequency, Q is thequality factor of the resonator, and κ₁ is the coupling coefficientbetween the counter-rotating modes introduced by the angular-momentumbiasing.

A three-port microstrip ring resonator was designed and simulated tooperate at approximately 885 MHz. Based on the design principles inEquation (4), the modulation frequency was set to 4.5 MHz. The resultsfrom the simulation confirmed the non-reciprocal response, with over 38dB isolation from the two output ports. The scattering parameters withand without angular-momentum biasing are shown in FIG. 6. FIG. 6 is agraph of the simulated results for the magnitude of the transmissioncoefficients in a microstrip resonator 601 designed following the sameprinciple as in FIG. 9 (discussed below) in accordance with anembodiment of the present invention. Without angular-modulation (AM)biasing, the transmission magnitude is the same in the two output ports(black line and marker), regardless of direction. When the modulationsignal is applied, the degenerate modes are split, leading to 38 dBisolation at 885 MHz. The physical layout of the ring resonator 601 withcoupling ports is shown in the inset of FIG. 6.

Referring to FIG. 6, in one embodiment, the RF signal is sent to ports1, 2 and 3 of ring resonator 601, the DC signal is sent to ports602A-602C and the modulation signal is sent to ports 603A-603C.

Without modulation, the transmission response is reciprocal, with themagnitude identical regardless of direction of propagation and reducedcompared to the modulated case, due to splitting of the power betweenthe two output ports. When the modulation is turned on, however, largeisolation and strong non-reciprocal response is achieved, withoutrelying on any magnetic material or bias.

Hence, the spatiotemporal azimuthal biasing method presented herein toproduce effective angular momentum in a resonant ring was shown toeffectively realize strong non-reciprocal response, without requiringferromagnetic materials or external magnetic bias. Passive, reciprocalresonators were experimentally shown to provide over 38 dB of isolationwhen the angular-momentum biasing technique was applied.

The angular-momentum-biasing concept is not only fully integratable inprinted circuit board technology, but it is also largely tunable andscalable; the same approach may be applied even to the terahertz andoptical spectra. The angular-momentum biasing method provides ascalable, cost effective, and compatible solution for conventional,non-reciprocal components. These results are believed to pave the way toa new route to replace conventional magnetically-biased non-reciprocalisolators, circulators and phase shifters with fully integratedcomponents.

In another embodiment of the present invention, non-reciprocity is basedon modulated coupled resonators thereby providing a different way toefficiently induce large non-reciprocity in a deeply subwavelengthdevice based on angular-momentum biasing. In this embodiment, thetrade-off between fabrication complexity and non-reciprocal effects isovercome. Furthermore, in this embodiment, non-reciprocity isdemonstrated with 100% modulation efficiency and only 3 separatemodulation regions. The proposed ring structure 700 is shown in FIG. 7A.FIG. 7A illustrates a ring structure 701 consisting of three stronglycoupled identical and symmetrically-coupled resonant tanks 701A-701Cwith resonance frequency A, and coupling factor A in accordance with anembodiment of the present invention. Resonant tanks 701A-701C maycollectively or individually be referred to as resonant tanks 701 orresonant tank 701, respectively. Modulation is applied here on theresonant frequencies of the individual tanks 701, so that they deviatefrom their static values as ω₁(t)=ω₀+δω_(m) cos(ω_(m)t), ω₂(t)=ω₀+δω_(m)cos(ω_(m)t+2π/3) and ω₃(t)=ω₀+δω_(m) cos(ω_(m)t+4π/3), with δω_(m) themodulation amplitude. Without modulation, such a device supports twodegenerate counter-rotating modes. When modulation is switched on,however, the degeneracy is lifted and non-reciprocity is produced. Theexperimental evidence provided herein is in the radio-frequency (RF)band, but this approach is applicable to any frequency band and even todifferent types of waves, since no assumption is made whatsoever aboutthe type of resonators.

In the absence of modulation (δω_(m)=0), the ring of FIG. 7A supportsthree states: a common one with state vector |c

=[1 1 1]^(T) and frequency ω_(c)=ω₀+2κ, and two degenerate right- andleft-handed ones with state vectors |±

=[1 e^(±i2π/3) e^(±i4π/3)]^(T) and frequencies ω₊=ω⁻=ω₀−κ. Thecomponents of the state vectors provide the complex amplitudes of thethree resonators. The applied modulation mixes right- and left-handedstates, producing two new hybrid states

$\begin{matrix}{{\left. {{\left. {\left. {\left. {{\left. {\left. \left| R \right. \right\rangle = \left| + \right.} \right\rangle{\mathbb{e}}^{{\mathbb{i}\omega}_{R}t}} - \frac{\Delta\omega}{{\delta\omega}_{m}}} \middle| - \right\rangle{\mathbb{e}}^{{- {{\mathbb{i}}{({\omega_{R} - \omega_{m}})}}}t}} \middle| L \right\rangle = \left| - \right.} \right\rangle{\mathbb{e}}^{{\mathbb{i}\omega}_{L}t}} + \frac{\Delta\omega}{{\delta\omega}_{m}}} \middle| + \right\rangle{\mathbb{e}}^{{- {{\mathbb{i}}{({\omega_{L} + \omega_{m}})}}}t}},} & (5)\end{matrix}$where Δω=√{square root over (ω_(m) ²+δω_(m) ²)}−ωm, ω_(R)=ω₊−Δω/2 andω_(L)=ω₊+Δω/2. Equation (5) is valid to the order

(δω_(m) ²) for ω_(m)<<ω_(c)−ω₊. Notice, that the hybrid states |R

and |L

have an expression identical to those of the ring of FIG. 1A. However,in contrast to its functionality, which requires a continuous,fast-oscillating spatiotemporal modulation, the structure of FIG. 7A ismodulated with only three signals.

For ω_(m)>0 the sub-states of each hybrid state have differentamplitudes, indicating the existence of a dominant and a secondarysub-state for each hybrid state: |+

and |−

are the dominant sub-states of |R

and |L

, respectively. The dominant sub-states support non-reciprocity, whilethe secondary ones result in intermodulation products at frequenciesω±ω_(m) for an incident signal with frequency ω. In a proper design,these by-products should fall outside the band of interest, so that theycan be easily filtered out. Such a requirement is satisfied if ω_(m) islarger than the bandwidth of the input signal, thus setting a lowerlimit on ω_(m) in practical applications. It can be seen from FIG. 7Bthat the power distributed to the secondary sub-states, i.e., theintermodulation product, decreases as ω_(m) increases, however with asimultaneous decrease of the frequency separation Δω between thedominant sub-states. A smaller Δω generally requires a higher Q-factorfor the coupled resonator, indicating a trade-off betweenintermodulation suppression and overall bandwidth. FIG. 7B is afrequency diagram of the hybrid states |R

and |L

of the modulated ring versus ω_(m) in accordance with an

embodiment of the present invention. The thickness of the linesrepresents the energy carried by the sub-states of the hybrid states.

The structure of FIG. 7A was realized at RF using three simple L-Ctanks, as in FIG. 8A, where the capacitance C is equally distributed atboth sides of the inductance L to maintain a symmetric structure. FIG.8A illustrates the constituent resonator 801 of the ring 700 (FIG. 7A):an L-C tank with modulated capacitance in accordance with an embodimentof the present invention.

Referring to FIG. 8A, capacitance is equally distributed at both sidesof the inductance for symmetry purposes. Capacitance modulation isachieved via varactor diodes 802, controlled by a static signal V_(dc)and the modulation signal ν_(m)(t). The resonance frequency modulationis achieved via capacitance modulation, commonly obtained in RF withvaractor diodes. These diodes are biased by two signals, a static oneV_(dc), which provides the required reverse bias to the diodes andcontrols their static capacitance, and an RF one ν_(m) with frequencyω_(m) and amplitude V_(m), providing the modulation. Assuming thatresonators 801 are coupled to each other through a capacitance C_(c), asin FIG. 8B, the frequencies of the common and rotating states readω_(c)=ω₀/√{square root over (1+2C_(c)/C)} and ω_(±)=ω₀√{square root over((C+3C_(c)/2)/(C+2C_(c)))}, respectively, where ω₀=1/√{square root over(LC)} is the resonance frequency of each tank. FIG. 8B illustrates aring 803 formed by three identical resonators 804A-804C coupled throughthree identical capacitances C_(c) 805A-805C in accordance with anembodiment of the present invention. Resonators 804A-804C maycollectively or individually be referred to as resonators 804 orresonator 804, respectively. Each resonator 804A-804C is configuredsimilarly as resonator 801 of FIG. 8A except being further coupled to anexternal microstrip transmission line 806A-806C through capacitances807A-807C, respectively. Capacitances 805A-805C may collectively orindividually be referred to as capacitances 805 or capacitance 805,respectively. Transmission lines 806A-806C may collectively orindividually be referred to as transmission lines 806 or transmissionline 806, respectively. Capacitances 807A-807C may collectively orindividually be referred to as capacitances 807 or capacitance 807,respectively.

Referring to FIG. 8B, if the amplitude of the capacitance modulation isΔC_(m), the frequency modulation amplitude is found asδω_(m)=ΔC_(m)/(2C). Note that ΔC_(m), and consequently δωm, areproportional to V_(m). Furthermore, it was observed that ω_(c), ω_(±)and δω_(m) are different from the predictions of the coupled-modeanalysis for the general structure of FIG. 7B, a known issue related tothe dependence of the average frequency of a coupled system on thecoupling element, which however, does not affect the validity ofEquation (5) and the subsequent analysis. As it will become clearer inthe following, ω_(c) should be as far as possible from ω_(±) in orderfor the common mode not to affect the operation of the structure atω_(±) where non-reciprocity occurs. In the lumped-element circuit ofFIG. 8B, this condition is satisfied by taking C_(c)→∞ or equivalentlycoupling the tanks through a short circuit, yielding ω_(c)=0 andω_(±)=ω₀√{square root over (3)}/2.

The non-reciprocal response of the circuit of FIG. 8B is demonstrated bycapacitively coupling it to three microstrip transmission lines 806, torealize a three-port device. Exciting the structure from, e.g., port 1,at frequency ω₊, results in the excitation of |R

and |L

with same amplitude and opposite phase φ_(R)=−φ_(L), due to thesymmetrical distribution of these states around ω₊. Then, the signals atports 2 and 3 are proportional to e^(i2π/3) e^(iφ) ^(R) +e^(i4π/3)e^(iφ)^(L) and e^(−i2π/3)e^(iφ) ^(R) +e^(−i4π/3)e^(iφ) ^(L) , respectively, asthe superposition of |R

and |L

at these ports. If δω_(m) and ω_(m) are selected so thatφ_(R)=−φ_(L)=−π/6, the signal at port 3 is identically zero, while theone at port 2 is non-zero, indicating routing of the incident power fromport 1 to port 2. Due to the symmetry of the structure with respect toits ports, incident power from ports 2 and 3 is similarly routed toports 3 and 1, thus realizing the functionality of a circulator withinfinite isolation. Notice, that the preceding description assumes aweak excitation of the common state, which makes clear the importance ofchoosing its resonance frequency as far as possible from the resonancefrequency of the rotating states.

FIG. 9 illustrates the physical layout of the RF non-reciprocalcoupled-resonator ring 800 in accordance with an embodiment of thepresent invention.

Referring to FIG. 9, ports 1, 2 and 3 provide access to ring 900 for theRF and modulation signals. Ports 4, 5 and 6 provide access to ring 900for the static biasing voltage. Elements 901-912 are effective for theRF signal; whereas, elements 913-924 are effective for the modulationsignal. A detail description of the circuit is provided below.

Ring 900 is designed to resonate at two frequencies: the modulationfrequency f_(m) and the radio frequency (RF) f_(RF). As a result, threeadditional ports are avoided for feeding the modulation signals andfilters that prevent the RF signal to leak into the modulation ports andvice-versa, thus significantly simplifying the design. The modulationand RF frequencies are respectively determined by elements 901-912 andelements 913-924 in FIG. 9, respectively. In particular,

$\begin{matrix}{\omega_{m} = {{2\pi\; f_{m}} = {\frac{2}{\sqrt{3}}\frac{1}{\sqrt{L_{1}C_{1}}}}}} & (6) \\{\omega_{RF} = {{2\pi\; f_{RF}} = {\frac{\sqrt{3}}{2}{\frac{1}{\sqrt{L_{2}C_{2}}}.}}}} & (7)\end{matrix}$

Equation (6) holds when the impedance of L₂ and L₁ are respectively muchsmaller than the corresponding of C₁ and C₂ at ω_(m). Similarly,Equation (7) holds when the impedance of L₂ and L₁ are respectively muchlarger than the corresponding of C₁ and C₂ at ω_(RF). These conditionsare summarized in the expressions

$\begin{matrix}{{{\omega_{m}L_{2}} ⪡ \frac{1}{\omega_{m}C_{1}}},{{\omega_{m}L_{1}} ⪡ \frac{1}{\omega_{m}C_{2}}},} & (8) \\{{{\omega_{RF}L_{2}} ⪢ \frac{1}{\omega_{RF}C_{1}}},{{\omega_{RF}L_{1}} ⪢ {\frac{1}{\omega_{RF}C_{2}}.}}} & (9)\end{matrix}$

Equation (8) implies that the inductors L₂ and the capacitors C₂ areshort and open circuits for the modulation signal at w. On the otherhand, Equation (9) implies that the capacitors C₁ and the inductors L₁are short and open circuits for the RF signal at ω_(RF). In practice,Equations (8) and (9) are considered to hold if

$\begin{matrix}{{{{\alpha\omega}_{m}L_{2}} = \frac{1}{\omega_{m}C_{1}}},{{{\alpha\omega}_{m}L_{1}} = \frac{1}{\omega_{m}C_{2}}},} & (10) \\{{{\omega_{RF}L_{2}} = \frac{\alpha}{\omega_{RF}C_{1}}},{{\omega_{RF}L_{1}} = \frac{\alpha}{\omega_{RF}C_{2}}},} & (11)\end{matrix}$where α≧10. Equations (10) and (11) are mutually satisfied ifω_(RF)=αω_(m). Then, solving Equations (6), (7), (10) and (11) yields

$\begin{matrix}{{L_{1} = {\frac{4\alpha}{3}L_{2}}},\mspace{14mu}{C_{1} = {\frac{4\alpha}{3}{C_{2}.}}}} & (12)\end{matrix}$

The capacitance C₂ is the static capacitance of the varactors. For thevaractor model used in the design of the present invention (Skyworks SMV1237) and for a static bias voltage of 3 V, C₂=30 pF. Then, iff_(RF)=150 MHz, L₂ is found from Equation (7) as L₂=28 nH. Subsequently,for the minimum value of α, α=10, f_(m)=15 MHz. L₁=370 nH and C₁=400 pF.

The capacitors C_(c) and the inductors L_(c) provide the coupling of theRF and modulation signals to ring 900. Similar to the resonant elementsof ring 900, the following conditions should hold in order for the RFand modulation signals to couple only through C_(c) and L_(c),respectively:

$\begin{matrix}{{{{\alpha\omega}_{RF}L_{c}} = \frac{1}{\omega_{RF}C_{c}}},} & (13) \\{{\omega_{m}L_{c}} = {\frac{\alpha}{\omega_{m}C_{c}}.}} & (14)\end{matrix}$

Observe that, as before, Equations (13) and (14) mutually hold ifω_(RF)=αω_(m). The coupling capacitance C_(c) determines the Q-factor ofring 900. For obtaining strong non-reciprocity, ω_(RF)˜Qω_(m), leadingto Q˜10, since ω_(RF)=10ω_(m). Having selected the value of C_(c) whichprovides the desired Q-factor, L_(c) can be calculated from Equation(14).

The inductors L_(rfc) are used to prevent the RF and modulation signalsfrom leaking to the DC source. For this purpose, any value larger than 1μm is sufficient. Indeed, the impedance of L_(rfc) at the RF frequencyof 150 MHz is 943Ω, which is much larger than the impedance 35Ω of thevaractor, meaning that the RF signal mainly flows through the varactor.Similarly, the impedance of L_(rfc) at the modulation frequency of 15MHz is 94Ω, which is fairly larger than the impedance 35Ω of theinductance L₁, meaning that the modulation signal mainly flows throughL₁. The capacitance C_(deb) blocks the DC signal from leaking to theground through L₁, while appearing as a short circuit at RF andmodulation frequencies. A value of 10 μF, corresponding to an impedanceof 1 mΩ and 0.1 mΩ at 15 MHz and 150 MHz, respectively, is enough forthis purpose.

The values of the lumped elements used in the fabricated layout arelisted in a table 1000 shown in FIG. 10 in accordance with an embodimentof the present invention. Notice, that these values are slightlydifferent than the ones calculated before due to restrictions in theavailable commercial elements. Elements 901-924 (FIG. 9) are of 0603 and0805 surface mount technology (SMT). Furthermore, the circuit wasfabricated in a FR4 substate and the external microstrip lines as wellas the ones connecting the elements between themselves were designed tohave a characteristic impedance of 50 Ω.

A description of the experimental setup is discussed below in connectionwith FIG. 11. FIG. 11 illustrates the experimental setup in accordancewith an embodiment of the present invention. As illustrated in FIG. 11in conjunction with FIG. 9, the static biasing of ring 900 is providedby a DC power supply 1101, whose output is connected to ports 4, 5 and 6of ring 900. For the generation of the modulation signals, a singlewaveform generator 1102 is used. The output of generator 1102 is splitevenly into three signals through a power divider 1103 and then routedto three phase shifters 1104A-1104C which provide the necessary phasedifference of 120° between the modulation signals. Phase shifters1104A-1104C are powered with a second DC source 1105 and potentiometers(not shown) are used to control their phase. The outputs of phaseshifters 1104A-1104C are connected to the low-pass ports 1106A-1106C ofthree diplexers 1107A-1107C in order to combine the modulation signalswith the RF ones. The high-pass ports 1108A, 1108C of two of thediplexers 1107A, 1107C are connected to the ports of a vector networkanalyzer 1109, while the high-pass port 1108B of diplexer 1107B isterminated to a matched load 1110. The outputs of diplexers 1107A-1107Care led to the ports 1, 2 and 3 of ring 900. Rotating diplexers1107A-1107C where the VNA ports are connected allows for the measurementof all the S-parameters of the circuit. The equipment used during themeasurement of ring 900 is listed in a table 1200 shown in FIG. 12 inaccordance with an embodiment of the present invention.

The realized device was designed to resonate at 170 MHz with a Q-factorof about 10 for V_(dc)=1.99 V and V_(m)=0. The modulation frequency wasset to 15 MHz, in order for the modulation by-products to fall outsidethe resonance band, whose bandwidth is here around 15 MHz. A photographof the fabricated prototype can be found in FIG. 8C in accordance withan embodiment of the present invention. Without modulation, the signalfrom each port is equally split to the other two ports, as expected fromsymmetry, and the system is fully reciprocal as illustrated in FIG. 13A.FIG. 13A is a graph illustrating the measured transmission from port 1to ports 2 and 3 for no modulation (V_(m)=0 V), where the power isequally split to the output ports (ports 2 and 3) as illustrated in theinset (representing circulator 803 of FIG. 8) in accordance with anembodiment of the present invention. The subscripts to the S-parametersor scattering parameters, as shown in FIG. 13A and in other Figures aswell as discussed below, represent the ports, where the first numberrepresents the final port and the second number represents the initialport in connection with the measured transmission.

When the modulation signal is switched on, this symmetry is broken andpower is unequally split to the two output ports. By varying themodulation amplitude, it is possible to find a value for which all thepower entering the ring from port 1 is routed to port 3, correspondingto the φ_(R)=−φ_(L)=−π/6 condition previously described. This is visiblein FIG. 13B, showing the measured S-parameters of the structure forV_(m)=0.6 V. FIG. 13B is a graph illustrating the measured scatteringparameters when V_(m)=0.6 V in accordance with an embodiment of thepresent invention. As illustrated in FIG. 13B, incident power to ports1, 2 and 3 is transmitted to ports 3, 1 and 2, respectively, thusrealizing a three-port circulator as illustrated in the inset(representing circulator 803). At the resonance frequency of 170 MHz,power incident to ports 1, 2 and 3 is routed to ports 2, 3 and 1,demonstrating the operation of an ideal, magnetic-free, deeplysubwavelength linear circulator.

For comparison, FIG. 13C presents the S-parameters obtained throughcombined full-wave and circuit simulations in accordance with anembodiment of the present invention. All the results in FIGS. 13A-13Crefer to V_(dc)=1.99 V.

In order to get deeper insights in the effect of V_(m) on the deviceoperation, FIG. 14A presents the transmission between ports 1 and 2 atresonance versus V_(m). FIG. 14A is a graph of the measured andsimulated transmission between ports 1 and 2 in accordance with anembodiment of the present invention. As illustrated in FIG. 14A,transmission is different for opposite propagation directions,indicating non-reciprocity. For V_(m)=0, S₂₁=S₁₂ as expected. IncreasingV_(m) results in a decrease of S₂₁ and an increase of S₁₂ untilV_(m)=0.6 V, where S₂₁=0. As mentioned above, this is the point whereφ_(R)=−φ_(L)=−π/6, i.e., where the counter-rotating modes interferedestructively at port 2. Past this point, S₂₁ and S₁₂ increase anddecrease, respectively, as expected when we depart from the destructiveinterference condition. For very large values of V_(m), S₂₁, and S₁₂tend to zero, since the counter-rotating states move far from each otherand, therefore, are weakly excited at ω₊. The magnitude of the asymmetrybetween S₂₁ and S₁₂ can be measured from the isolation S₁₂/S₂₁, plottedin FIG. 14B in logarithmic scale versus V_(m). FIG. 14B is a graph ofthe isolation (S₁₂/S₂₁) in a logarithmic scale in accordance with anembodiment of the present invention. As illustrated in FIG. 14B, for theoptimum modulation voltage V_(m)=0.6 V, S₁₂ is over 4 orders ofmagnitude larger than S₂₁, indicating extremely strong non-reciprocity,above the levels of commercial magnetic-based devices. All the resultsin FIGS. 14A-14B refer to V_(dc)=1.99 V.

Another key property of this realized device is its unique real-timetunability features. The biasing voltage V_(dc), which provides thenecessary reverse biasing condition for the operation of the varactordiodes, determines their static capacitance. Therefore, V_(dc) can beused to actively control the static resonance frequency of the L-Ctanks, and consequently the band over which non-reciprocity occurs. FIG.15 shows the measured isolation versus frequency for V_(dc) variedbetween 1.73 V and 4.5 V in a logarithmic scale in accordance with anembodiment of the present invention. Referring to FIG. 15, in each case,V_(m) is appropriately adjusted so that isolation at resonance becomesmaximum. It is clear that the non-reciprocal response of the device ofthe present invention can be efficiently tuned between 150 MHz and 210MHz, corresponding to a relative bandwidth of over 30%. Across all thisrange, the measured isolation is above 40 dB, and it even reaches 60 dBfor V_(dc)=4.5 V. This strong tuning capability is an additionaladvantage of the device compared to conventional magnetic-basedmicrowave circulators, and it may be exploited in scenarios requiringdynamic tuning to balance changes in temperature or in the environment.

In addition to being an ideal replacement for conventional microwavenon-reciprocal components, with significant advantages in terms of size,integration, cost, linearity and noise reduction, the findings presentedherein become particularly relevant if translated to integratednanophotonic technology, for which optical non-reciprocal components maybecome of crucial importance for laser protection and signal routing.The approach discussed herein is applicable to any frequency of theelectromagnetic spectrum, and therefore also to light as discussedfurther below in connection with FIG. 16. At such frequencies,electro-optic modulation in silicon-based components is typicallyachieved via carrier injection/depletion. Although such technology canprovide quite strong permittivity modulation, it is accompanied bysignificant loss, and low modulation frequencies for large modulationamplitudes. These side effects impose severe limitations on theapplicability of the principle of angular-momentum biasing in itsoriginal form based on uniform micro-ring resonators. As previouslyexplained, the strength of angular-momentum-based non-reciprocity iscontingent upon the quality factor of the ring resonators and theirmodulation frequency. Micro-ring resonators exhibit poor modulationefficiency, and, as a result, they require strong modulation. Strongmodulation in turn typically degrades the quality factor and limits themodulation frequency, thus also restricting the magnitude of theattainable non-reciprocity. On the other hand, the concept presentedhere ensures a modulation efficiency of 100%, significantly relaxing therequirements in terms of the modulation amplitude. This in turn allowslarge quality factors and large modulation frequencies, which directlytranslate into strong non-reciprocal response in deeply subwavelengthdevices. Furthermore, the introduced approach can be readily applied tophotonic crystal technology, for which efficient high-Q coupled cavitiesmay be easily implemented and efficiently modulated.

Referring to FIG. 16, FIG. 16 illustrates a photonic crystal 1600 usedfor building resonators for light in accordance with an embodiment ofthe present invention. Photonic crystal 1600 includes an array ofdielectric rods 1601, which prevents light propagation in a particularfrequency range or photonic bandgap. Photonic crystal 1601 furtherincludes defects or gaps 1602A-1602C, such as three of them as shown inFIG. 6, between these dielectric rods 1601. Light is localized (i.e.light cannot escape) at these gaps 1602A-1602C for frequencies in thephotonic bandgap of the array of dielectric rods. In this manner, gaps1602A-1602C can function as a resonator. Gaps 1602A-1602C can bemodulated via electrical or optical means. Gaps 1602A-1602C are alsocoupled to three external ports through three waveguides formed byremoving three series of dielectric rods in the photonic crystal.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

The invention claimed is:
 1. A non-reciprocal device comprising: a ringof three or more resonators coupled together, wherein each of said threeor more resonators is modulated in time with a different phase shiftwith respect to each other and operable at either microwave, light orsound waves.
 2. The non-reciprocal device as recited in claim 1, whereineach of said three or more resonators comprises an LC circuit.
 3. Thenon-reciprocal device as recited in claim 1, wherein said ring iscoupled to two or more external ports.
 4. The non-reciprocal device asrecited in claim 3, wherein said two or more external ports areconfigured to provide access to radio frequency and modulation signals.5. The non-reciprocal device as recited in claim 4, wherein saidmodulation signals are generated by a waveform generator, wherein anoutput of said generator is split evenly into three or more signalswhich are phase shifted with respect to each other via phase shifters.6. The non-reciprocal device as recited in claim 1, wherein said threeor more resonators are coupled together via capacitances.
 7. Thenon-reciprocal device as recited in claim 1, wherein each of said threeor more resonators is modulated via capacitance modulation, wherein avaractor diode is configured to provide said capacitance modulation. 8.The non-reciprocal device as recited in claim 7, wherein said varactordiode is biased by a direct current signal providing a reverse bias tosaid varactor diode to actively control a static resonance frequency ofsaid three or more resonators.
 9. The non-reciprocal device as recitedin claim 7, wherein said varactor diode is biased by a radio frequencysignal providing said modulation.